Existence and uniqueness conjecture for submanifolds of nearly Kähler $\mathbb{C}P^3$

Let $(M,g)$ be an $n$-dimensional simply connected Riemannian manifold with $n<6$, and let $\mathcal{E}$ be a Riemannian vector bundle of rank $p<6$ over $M$, equipped with a metric-compatible connection $\nabla^\mathcal{E}$ and curvature tensor $R^\mathcal{E}$. Let $h^\mathcal{E}:TM\times TM\to\mathcal{E}$ be smooth and symmetric, and define $A^\mathcal{E}_\xi$ for each section $\xi$ of $\mathcal{E}$ by

$$g(A^\mathcal{E}_\xi X,Y)=g(h^\mathcal{E}(X,Y),\xi).$$

Assume that $(\nabla^\mathcal{E},h^\mathcal{E},A^\mathcal{E},R^\mathcal{E})$ satisfy the Gauss, Codazzi and Ricci equations for the nearly Kähler $\mathbb{C}P^3$, and that

$$(\nabla J_\circ)(X,Y)=\frac{J+J_\circ}{2}G\left(\frac{J-J_\circ}{2}X,Y\right).$$

Existence and uniqueness conjecture. There exists an isometric immersion $f:M\to(\mathbb{C}P^3,g,J)$ and a vector bundle isometry $\varphi:\mathcal{E}\to T^\perp M$ such that $\nabla^\perp\varphi=\varphi\nabla^\mathcal{E}$ and $h=\varphi\circ h^\mathcal{E}$. Moreover, if $f_1,f_2:M\to(\mathbb{C}P^3,g,J)$ are isometric immersions and there is a vector bundle isometry $\varphi:T^{\perp,f_1}M\to T^{\perp,f_2}M$ satisfying $\varphi\circ\nabla^{\perp,f_1}=\nabla^{\perp,f_2}\circ\varphi$ and $\varphi\circ h^{f_1}=h^{f_2}$, then the immersions differ by an ambient isometry whose differential restricts to $\varphi$ on the normal bundle. The fundamental equations together with the additional equation above provide existence and uniqueness in all examples considered, but a general result has not been proved; this conjecture would establish a general fundamental theorem for submanifolds of nearly Kähler $\mathbb{C}P^3$.